1. Field of the Invention
The present invention relates to a characteristic evaluation method for insulated gate type transistors which extracts their effective channel widths, a characteristic evaluation apparatus for insulated gate type transistors, a method of manufacturing insulated gate type transistors by using the above characteristic evaluation method, and a computer readable storing medium storing a characteristic evaluation program.
2. Description of the Background Art
An electrically effective channel width, i.e., an effective channel width Weff, can be determined from the drain currents of two or more insulated gate type transistors having the same channel length and a different channel width. This method is generally called xe2x80x9cdrain current method.xe2x80x9d The drain current method can directly determine the difference between an effective channel width Weff and a mask channel width Wm, namely, a channel narrowing DW(=Wmxe2x88x92Weff).
As a drain current method, a wide variety of methods have been proposed heretofore. They are described, for example, in xe2x80x9cA New Method to Electrically determine Effective MOSFET Channel Widthxe2x80x9d by Y. R. Ma and K. L. Wang, IEEE Trans. Elect. Dev., ED-29, p. 1825, 1982; xe2x80x9cA New Method to Determine the MOSFET Effective Channel Widthxe2x80x9d by N. D. Arora, L. A. Blair and L. M. Richardson, IEEE Trans. Elect. Dev., ED-37(3), p. 811, 1990; xe2x80x9cA Method to Extract Gate-Bias-Dependent MOSFET""s Effective Channel Widthxe2x80x9d by Y. T. Chia and G. J. Hu, IEEE Trans. Elect. Dev., ED-38(2), p. 424, 1991; and xe2x80x9cA Direct Method to Extract Effective Geometries and Series Resistances of MOS Transistorsxe2x80x9d by P. R. Karlsson and K. O. Jeppson, Proc. IEEE ICMTS, vol. 7, p. 184, 1994.
Of various drain current methods, Chia method is commonly often used. Thus, Chia method will be briefly described here. The total source-drain resistance R is given by the sum of a channel resistance Rch and an external resistance Rsd. Now, supposing the following Equation 1 as the equation to express drain current.                               I          ds                =                                            β              0                        ·                          (                                                V                  gs                                -                                  V                  th                                -                                                      V                    ds                    *                                    2                                            )                        ·                          V              ds              *                                            1            +                          θ              ⁢                              xe2x80x83                            ⁢                              1                ·                                  (                                                            V                      gs                      *                                        -                                          V                      th                                                        )                                                      +                          θ              ⁢                              xe2x80x83                            ⁢                              2                ·                                                      (                                                                  V                        gs                        *                                            -                                              V                        th                                                              )                                    2                                                                                        (                  Eq          .                      xe2x80x83                    ⁢          1                )            
where xcex20, Vds* and Vgs* are given by the following Equations 2, 3 and 4, respectively, and xcex81 and xcex82 are the invariables.                               β          0                =                                            μ              0                        ⁢                          C              ox                        ⁢                          W              eff                                            L            eff                                              (                  Eq          .                      xe2x80x83                    ⁢          2                )            
where xcexc0 is a carrier mobility, Leff is an effective channel length, Weff is an effective channel width, and Cox is a gate insulating film capacity.
Vds*=Vdsxe2x88x92Idsxc2x7Rsdxe2x80x83xe2x80x83(Eq. 3)
                              V          gs          *                =                              V            gs                    -                                                    I                ds                            ·                              R                sd                                      2                                              (                  Eq          .                      xe2x80x83                    ⁢          4                )            
Neglecting the term of xcex82, Equation 5 is obtained from Equations 1, 3 and 4. Supposing an external resistance Rsd is inversely proportional to an effective channel width Weff, a channel narrowing DW can be extracted through the following procedure.                               I          ds                =                                            β              0                        ·                          (                                                V                  gs                                -                                  V                  th                                -                                                      V                    ds                                    2                                            )                        ·                          V              ds                                            1            +                                          (                                                      θ                    ⁢                                          xe2x80x83                                        ⁢                    1                                    +                                                            β                      0                                        ·                                          R                      sd                                                                      )                            ·                              (                                                      V                    gs                                    -                                      V                    th                                                  )                                                                        (                  Eq          .                      xe2x80x83                    ⁢          5                )            
where the difference between a gate voltage and a threshold voltage, (Vgsxe2x88x92Vth), is defined as a gate overdrive Vgt.
Step 1: Against a certain gate overdrive Vgt, Idsxe2x88x92Wm characteristic is plotted in an X-Y plane whose X-axis is mask channel Wm and Y-axis is drain current Ids, and a linear fitting is made. At that time, the intersection with the X-axis in the X-Y plane which is obtained by extrapolating each straight line is the channel narrowing DW (Vgt) in the gate overdrive Vgt, (see FIG. 1).
Step 2: By repeating step 1 while changing the gate overdrive Vgt, it can be seen how the channel narrowing DW (Vgt) depends on the gate overdrive Vgt (see FIG. 1).
Prior art characteristic evaluation method for insulated type transistors is constructed as described. In Chia method, for example, it is necessary to know the threshold voltage of a transistor for use in extraction. The threshold voltage of a transistor is found by, for example, extrapolation from the characteristic between gate voltage and source-drain current, as shown in FIG. 2. Therefore, the error due to the uncertainty of a threshold voltage is further pronounced with reducing transistor size.
According to a first aspect of the present invention, a characteristic evaluation apparatus for insulated gate type transistors in which at least two insulated gate type transistors that differ from each other only in mask channel width are used for evaluation and the characteristic of a first insulated gate type transistor having a wide mask channel width serves as a reference, to evaluate the characteristic of a second insulated gate type transistor having a narrow mask channel width. This apparatus comprises: a threshold voltage estimation means that extracts the threshold voltage of the first transistor, estimates the threshold voltage of the second transistor, and employs a value as estimated, as a first estimated value; an extraction means in which (i) a difference between a gate voltage of the first transistor and the extracted threshold voltage of the first transistor is defined as a first gate overdrive, and a difference between a gate voltage of the second transistors and the first estimated value is defined as a second gate overdrive, (ii) in an X-Y plane whose X-axis is the mask channel width and Y-axis is source-drain conductance, a virtual point at which a change of Y coordinate value is estimated to be approximately zero when the first and second gate overdrives are finely changed, is extracted from a characteristic curve exhibiting a relationship between the mask channel widths of the first and second transistors and the source-drain conductance, (iii) values of the X coordinate and Y coordinate at the virtual point are defined as second and third estimated values, respectively, and (iv) a slope of the characteristic curve at the virtual point is extracted and a value of the slope is employed as a fourth estimated value; a threshold voltage determination means in which (i) from the second to fourth estimated values, optimum second to fourth estimated values are found with which the change of the third estimated value is equal to the product of the change of the second estimated value and the fourth estimated value, in reply to fine changes of the first and second gate overdrives, (ii) an optimum first estimated value is determined which corresponds to the optimum second to fourth estimated values, and (iii) a true threshold voltage of the second transistor is determined based on the optimum first estimated value; and a channel narrowing determination means that determines a difference between the mask channel width and an effective channel width, based on the true threshold voltage.
According to a second aspect, the characteristic evaluation apparatus of the first aspect is characterized in that the extraction means approximates the characteristic curve by using a first straight line in the X-Y plane, the first straight line passing through a first point that is given to the first transistor when the first gate overdrive has a first value and a second point that is given to the second transistor when the second gate overdrive has the first value.
According to a third aspect, the characteristic evaluation apparatus of the second aspect is characterized in that the threshold voltage determination means determines the optimum second to fourth estimated values from a relational expression:       F    ⁡          (              δ        ,                  V          gtWi                    )        =                    dW        **            ⁡              (                  δ          ,                      V            gtWi                          )              +                            f          ⁡                      (                          δ              ,                              V                gtWi                                      )                                                f            xe2x80x2                    ⁡                      (                          δ              ,                              V                gtWi                                      )                              ·                        dW                      **            '                          ⁡                  (                      δ            ,                          V              gtWi                                )                      -                  DW        *            ⁡              (                  δ          ,                      V            gtWi                          )            
where xcex4 is a difference between an estimated value of the threshold voltage of the second transistor, i.e., a first estimated value, and the threshold voltage of the first transistor; VgtWi is the first gate overdrive; dW** is a value of an X intercept that is obtained by extrapolating the characteristic curve; f is the slope of the characteristic curve at the virtual point; DW* is an X coordinate value at the virtual point; and a prime is the first-order differentiation of VgtWi.
According to a fourth aspect, the characteristic evaluation apparatus of the second aspect is characterized in that the threshold voltage determination means determines the optimum second to fourth estimated values from a relational expression:       F    ⁡          (              δ        ,                  V          gtWi                    )        =                                          f            2                    ⁡                      (                          δ              ,                              V                gtWi                                      )                                                f            xe2x80x2                    ⁡                      (                          δ              ,                              V                gtWi                                      )                              ·                        dW                      **            '                          ⁡                  (                      δ            ,                          V              gtWi                                )                      -                  G        m        *            ⁡              (                  δ          ,                      V            gtWi                          )            
where xcex4 is a difference between an estimated value of the threshold voltage of the second transistor, i.e., a first estimated value, and the threshold voltage of the first transistor; VgtWi is the first gate overdrive; dW** is a value of an X intercept that is obtained by extrapolating the characteristic curve; f is the slope of the characteristic curve at the virtual point; Gm* is a Y coordinate value at the virtual point; and a prime is the first-order differentiation of VgtWi.
According to a fifth aspect, the characteristic evaluation apparatus of the second aspect is characterized in that the threshold voltage determination means determines the optimum second to fourth estimated values from a relational expression:       F    ⁢          (              δ        ,                  V          gtWi                    )        =                    G        m        **            ⁡              (                  δ          ,                      V            gtWi                          )              -                            f          ⁡                      (                          δ              ,                              V                gtWi                                      )                                                f            xe2x80x2                    ⁡                      (                          δ              ,                              V                gtWi                                      )                              ·                        G          m                      **            '                          ⁡                  (                      δ            ,                          V              gtWi                                )                      -                  G        m        *            ⁡              (                  δ          ,                      V            gtWi                          )            
where xcex4 is a difference between an estimated value of the threshold voltage of the second transistor, i.e., a first estimated value, and the threshold voltage of the first transistor; VgtWi is the first gate overdrive; Gm** is a value of a Y intercept that is obtained by extrapolating the characteristic curve; f is the slope of the characteristic curve at the virtual point; Gm* is a Y coordinate value at the virtual point; and a prime is the first-order differentiation of VgtWi.
According to a sixth aspect, the characteristic evaluation apparatus of the second aspect is characterized in that the threshold voltage determination means determines the optimum second to fourth estimated values from a relational expression:       F    ⁡          (              δ        ,                  V          gtWi                    )        =                              G          m                      **            '                          ⁡                  (                      δ            ,                          V              gtWi                                )                                      f          xe2x80x2                ⁡                  (                      δ            ,                          V              gtWi                                )                      +                  DW        *            ⁡              (                  δ          ,                      V            gtWi                          )            
where xcex4 is a difference between an estimated value of the threshold voltage of the second transistor, i.e., a first estimated value, and the threshold voltage of the first transistor; VgtWi is the first gate overdrive; Gm** is a value of a Y intercept that is obtained by extrapolating the characteristic curve; f is the slope of the characteristic curve at the virtual point; DW* is an X coordinate value at the virtual point; and a prime is the first-order differentiation of VgtWi.
According to a seventh aspect, a characteristic evaluation apparatus for insulated gate type transistors in which at least two insulated gate type transistors that differ from each other only in mask channel width are used for evaluation and the characteristic of a first insulated gate type transistor having a wide mask channel width serves as a reference, to evaluate the characteristic of a second insulated gate type transistor having a narrow mask channel width. This apparatus comprises: a threshold voltage estimation means that extracts the threshold voltage of the first transistor, estimates the threshold voltage of the second transistor, and employs a value as estimated, as a first estimated value; an extraction means in which (i) a difference between a gate voltage of the first transistor and the threshold voltage of the first transistor is defined as a first gate overdrive, and a difference between a gate voltage of the second transistor and the first estimated value is defined as a second gate overdrive, (ii) in an X-Y plane whose X-axis is the mask channel width and Y-axis is source-drain conductance, a virtual point at which a change in Y coordinate value is estimated to be approximately zero when the first and second gate overdrives are finely changed from a first characteristic curve exhibiting a relationship between the mask channel widths of the first and second transistors and the source-drain conductance, and (iii) a value of the X coordinate at the virtual point is employed as a second estimated value, alternatively, as a value of the X intercept of the first characteristic curve; a threshold voltage determination means in which (i) from the second estimated value, an optimum first estimated value is found with which a second characteristic curve exhibiting a relationship between the second gate overdrive and the second estimated value in an X-Y plane whose X-axis is the second gate overdrive and Y-axis is a value related to the second estimated value, has a predetermined shape within a predetermined range of the second gate overdrive, and (ii) the optimum first estimated value is determined as a true threshold voltage of the second transistor; and a channel narrowing determination means that determines a difference between the mask channel width and an effective channel width, based on the true threshold voltage.
According to an eighth aspect, the characteristic evaluation apparatus of the seventh aspect is characterized in that the extraction means further employs a value of the X intercept of the first characteristic curve as a third estimated value; and the threshold voltage determination means employs a value that is obtained by reducing the second estimated value from twice the third estimated value, as the value related to the second estimated value.
According to a ninth aspect, the characteristic evaluation apparatus of the eighth aspect is characterized in that the threshold voltage determination means employs the first estimated value with which a value that is obtained by reducing the second estimated value from twice the third estimated value is best converged on a fixed value in the predetermined range, as the optimum first estimated value.
According to a tenth aspect, the characteristic evaluation apparatus of the first aspect is characterized in that the channel narrowing determination means determines a difference between the mask channel width and an effective channel width, from a value that is obtained by reducing the second estimated value from twice the third estimated value when the gate overdrive is in the vicinity of 0 V.
According to an eleventh aspect, a characteristic evaluation apparatus for insulated gate type transistors in which at least two insulated gate type transistors that differ from each other only in mask channel width are used for evaluation and the characteristic of a first insulated gate type transistor having a wide mask channel width serves as a reference, to evaluate the characteristic of a second insulated gate type transistor having a narrow mask channel width. This apparatus comprises: a threshold voltage estimation means that extracts a threshold voltage of the first transistor, estimates the threshold voltage of the second transistor, and employs a value as estimated, as a first estimated value; an extraction means in which (i) a difference between a gate voltage of the first transistor and the extracted threshold voltage of the first transistor is defined as a first gate overdrive, and a difference between a gate voltage of the second transistor and the first estimated value is defined as a second gate overdrive, (ii) under the condition that the first and second gate overdrives are the same in an X-Y plane whose X-axis is the mask channel width and Y-axis is source-drain resistance, a virtual point at which a change in Y coordinate value is estimated to be approximately zero even if the first and second gate overdrives are finely changed, is extracted from points on a straight line passing through a first point whose X coordinate is the mask channel width of the first transistor and Y coordinate is the source-drain resistance of the second transistor, and a second point whose X coordinate is the mask channel width of the second transistor and Y coordinate is the source-drain resistance of the first transistor, (iii) values of the X coordinate and Y coordinate at the virtual points are defined as second and third estimated values, respectively, and (iv) a slope of the straight line at the virtual points is extracted and a value of the slope is employed as a fourth estimated value; a threshold voltage determination means that determines a true threshold voltage of the second transistor by using the first to fourth estimated values; and a channel narrowing determination means that determines a difference between the mask channel width and an effective channel width, based on the true threshold voltage.
According to a twelfth aspect, in the characteristic evaluation apparatus of the eleventh aspect the threshold voltage determination means is characterized in: (i) finding, from the second to fourth estimated values, optimum second to fourth estimated values with which a change of the third estimated value is equal to the product of a change of the second estimated value and the fourth estimated value, in reply to fine changes of the first and second gate overdrives, (ii) determining an optimum first estimated value that corresponds to the optimum second to fourth estimated values, and (iii) determining the true threshold voltage of the second transistor, based on the optimum first estimated value.
According to a thirteenth aspect, the characteristic evaluation apparatus of the twelfth aspect is characterized in that the threshold voltage determination means determines the optimum second to fourth estimated values from a relational expression:       F    ⁡          (              δ        ,                  V          gtWi                    )        =                                          h            2                    ⁡                      (                          δ              ,                              V                gtWi                                      )                                                h            xe2x80x2                    ⁡                      (                          δ              ,                              V                gtWi                                      )                              ·                        dW                      **            '                          ⁡                  (                      δ            ,                          V              gtWi                                )                      -                  R        #            ⁡              (                  δ          ,                      V            gtWi                          )            
where xcex4 is a difference between an estimated value of the threshold voltage of the second transistor, i.e., a first estimated value, and the threshold voltage of the first transistor; VgtWi is the first gate overdrive; dW** is a value of an X intercept that is obtained by extrapolating the straight line; h is the slope of the straight line; R# is a Y coordinate value at the virtual point; and a prime is the first-order differentiation of VgtWi.
According to a fourteenth aspect, the characteristic evaluation apparatus of the twelfth aspect is characterized in that the threshold voltage determination means determines the optimum second to fourth estimated values from a relational expression:       F    ⁡          (              δ        ,                  V          gtWi                    )        =                    R        **            ⁡              (                  δ          ,                      V            gtWi                          )              -                            h          ⁡                      (                          δ              ,                              V                gtWi                                      )                                                h            xe2x80x2                    ⁡                      (                          δ              ,                              V                gtWi                                      )                              ·                        R                      **            '                          ⁡                  (                      δ            ,                          V              gtWi                                )                      -                  R        #            ⁡              (                  δ          ,                      V            gtWi                          )            
where xcex4 is a difference between an estimated value of the threshold voltage of the second transistor, i.e., a first estimated value, and the threshold voltage of the first transistor; VgtWi is the first gate overdrive; R** is a value of a Y intercept that is obtained by extrapolating the straight line; h is the slope of the straight line; R# is a Y coordinate value at the virtual point; and a prime is the first-order differentiation of VgtWi.
According to a fifteenth aspect, the characteristic evaluation apparatus of the twelfth aspect is characterized in that the threshold voltage determination means determines the optimum second to fourth estimated values from a relational expression:       F    ⁡          (              δ        ,                  V          gtWi                    )        =                              R                      **            '                          ⁡                  (                      δ            ,                          V              gtWi                                )                                      h          xe2x80x2                ⁡                  (                      δ            ,                          V              gtWi                                )                      +                  DW        #            ⁡              (                  δ          ,                      V            gtWi                          )            
where xcex4 is a difference between an estimated value of the threshold voltage of the second transistor, i.e., a first estimated value, and the threshold voltage of the first transistor; VgtWi is the first gate overdrive; R** is a value of a Y intercept that is obtained by extrapolating the straight line; h is the slope of the straight line; DW# is an X coordinate value at the virtual point; and a prime is the first-order differentiation of VgtWi.
According to a sixteenth aspect, the characteristic evaluation apparatus of the twelfth aspect is characterized in that the threshold voltage determination means determines the optimum second to fourth estimated values from a relational expression:       F    ⁡          (              δ        ,                  V          gtWi                    )        =                    dW        **            ⁡              (                  δ          ,                      V            gtWi                          )              +                            h          ⁡                      (                          δ              ,                              V                gtWi                                      )                                                h            xe2x80x2                    ⁡                      (                          δ              ,                              V                gtWi                                      )                              ·                        dW                      **            '                          ⁡                  (                      δ            ,                          V              gtWi                                )                      -                  DW        #            ⁡              (                  δ          ,                      V            gtWi                          )            
where xcex4 is a difference between an estimated value of the threshold voltage of the second transistor, i.e., a first estimated value, and the threshold voltage of the first transistor, VgtWi is the first gate overdrive; dW** is a value of an X intercept that is obtained by extrapolating the straight line; h is the slope of the straight line; DW# is an X coordinate value at the virtual point; and a prime is a first-order differentiation of VgtWi.
According to a seventeenth aspect, in the characteristic evaluation apparatus of the eleventh aspect the threshold voltage determination means is characterized in (i) finding, in an X-Y plane whose X-axis is the second gate overdrive and Y-axis is the second estimated value, the optimum first estimated value with which a characteristic curve exhibiting the relationship between the second gate overdrive and the second estimated value has a predetermined shape in a predetermined range of the second gate overdrive, and (ii) determining the true threshold voltage of the second transistor, based on the optimum first estimated value.
According to an eighteenth aspect, the characteristic evaluation apparatus of the seventeenth aspect is characterized in that the threshold voltage determination means estimates, from the characteristic curve in plural, an optimum characteristic curve with which the second estimated value is best converged on a fixed value in the predetermined range.
According to a nineteenth aspect, the characteristic evaluation apparatus of the eleventh aspect is characterized in that the channel narrowing determination means determines a difference between the mask channel width and an effective channel width, from the second estimated value when the gate overdrive is in the vicinity of 0 V.
The characteristic evaluation apparatus of the first or twelfth aspect allows accurate extraction of the threshold voltage of the second insulated gate type transistor, irrespective of the range of the second gate overdrive, thereby improving the accuracy of effective channel width extraction.
The characteristic evaluation apparatus of the eleventh aspect facilitates to determine the value of channel narrowing when the first and second gate overdrives are in the vicinity of zero because the stationary point of the second estimated value is present in the vicinity of zero.
The characteristic evaluation apparatus of the second aspect facilitates the slope extraction between virtual points because a characteristic curve is approximated to a straight line. This allows to find a virtual point as the intersection of straight lines, and the slope at an intersection as the slope of a straight line.
The characteristic evaluation apparatus of the third, fourth, fifth, sixth, thirteenth, fourteenth, fifteenth or sixteenth aspect requires no differentiation of the gate overdrive at a virtual point, thereby reducing errors.
The characteristic evaluation apparatus of the seventh, eighth or seventeenth aspect facilitates to determine true threshold voltages because the second characteristic curves that are obtained for the true threshold voltage on a graph may approximately coincide, irrespective of mask channel width.
The characteristic evaluation apparatus of the ninth or eighteenth aspect facilitates programming for appropriate results by detecting an optimum characteristic curve exhibiting the best convergence on a fixed value.
The characteristic evaluation apparatus of the tenth or nineteenth aspect facilitates channel narrowing determination because the channel narrowing at the gate overdrive of 0 V is determined by using a value that is obtained by reducing the second estimated value from twice the third estimated value, alternatively, because the second estimated value has a stationary point when the gate overdrive is in the vicinity of 0 V.
To solve the above problem, it is an object of the present invention to obtain a characteristic evaluation apparatus for insulating gate type transistors which performs evaluation of insulated gate type transistors by using a characteristic evaluation method for insulated gate type transistors which reduces the error due to the uncertainty of a threshold voltage to permit channel narrowing extraction of high accuracy.
It is another object of the present invention to obtain a computer readable storing medium that stores a characteristic evaluation program.
It is another object of the present invention to obtain a manufacturing method by which insulated gate type transistors having excellent characteristics can be manufactured easily by using the above characteristic evaluation method.